The free strict monoidal category with k monoids

It is well known that the augmented simplex category is the free strict monoidal category with a monoid: the monoidal product is concatenation, and the monoid is the terminal object (called [0] or Δ0 by geometers). Perhaps less well known is the free strict monoidal category with k monoids (where k is any cardinal, possibly infinite):

  • The objects are pairs (n, P) where n is a natural number (possibly zero) and P is a partition of the set { 0, …, n – 1 } into k disjoint subsets Pj, 0 ≤ j < k.
  • The morphisms (n, P) → (m, Q) are monotone maps { 0, …, n – 1 } → { 0, …, m – 1 } that respect the partitions, i.e. send members of Pj to members of Qj.

In other words, it is the category of finite linear orders equipped k-colouring of its elements. From this point of view, the monoidal product is best interpreted as concatenation of (coloured) linear orders. The k monoids are the k different objects of the form (1, P). Amusingly, the monoidal product on the set of objects makes it the free monoid on k generators.

The opposite category, i.e. the free strict monoidal category with k comonoids, can be described concretely as follows:

  • The objects are pairs (n, P) as before.
  • The morphisms (m, Q) → (n, P) are monotone maps f : { 0, …, m } → { 0, …, n } that satisfy the following axioms:
    • f(0) = 0 and f(m) = n.
    • If i is a member of Pj, then for f(i) ≤ e < f(i + 1), e is a member of Qj .

The above should be interpreted as the category of finite intervals equipped with a k-colouring of its edges. From this point of view, the monoidal product is the wedge sum. The special case where k = 2 is especially useful in relation to the problem of inverting morphisms in a category: for this, see [Dwyer and Kan, Calculating simplicial localizations].

The quantale of ideals

It had long puzzled me why people usually use the ideal product to show that the intersection of any two Zariski-open subsets of the prime spectrum of a ring is again a Zariski-open subset – after all, the intersection of the two ideals would do just as well. But perhaps this is the reason:

Proposition. Let f : AB be a ring homomorphism and let I and J be ideals of A. Then the ideal of B generated by f(IJ) is the ideal product of the ideals generated by fI and fJ.

It is well known that the poset of ideals of a ring has arbitrary joins (namely, the ideal sum) and that they are preserved by pushforward along ring homomorphisms. Moreover, the ideal product distributes over the ideal sum, so the poset of ideals of a ring in fact a quantale, and we have a functor CRingQuant. A result of Mulvey says that the frame of Zariski-open subsets of the prime spectrum of A is the frame reflection of the quantale of ideals of A.

While it is true that the poset of ideals is a complete lattice, not all binary meets (i.e. intersections) are preserved by pushforward. (Consider the homomorphism k[x, y] / (xy) → k[z] / (z2) sending x and y to z.) Distributivity also fails. (Consider the ring k[x, y] / (x2, xy, y).) Still, one might ask, what is the relationship between the frame of Zariski-open subsets of the prime spectrum and the frame reflection of the poset of ideals, considered as a join semilattice with finite meets?

A characterisation of local schemes

Let X be a scheme. The following are equivalent:

  1. X is a local scheme, i.e. of the form Spec A where A is a local ring.
  2. X is a scheme with a unique closed point x and the only (open) neighbourhood of x is X itself.
  3. Every open covering of X must contain X itself.

It is well known that the first condition implies the second, and it is an easy exercise (in point set topology) to show that the second condition implies the third. To complete the proof of the claim, it is enough to show that the third condition implies the first.

Suppose X is a scheme satisfying the third condition. Any scheme can be covered by open affine subschemes, so the condition on X implies it is affine. In particular, it has a closed point x. The condition also implies that the complement of x is the unique maximal open proper subset of X, so x must be the unique closed point of X, as required.

We should observe that the third condition can be expressed purely in terms of the Zariski topology on the category of schemes: it says that a scheme X is local if and only if every Zariski-covering family of X contains a split epimorphism, or equivalently, if and only if there is a unique Zariski-covering sieve on X (namely, the maximal sieve).

The standard model structure on Cat is canonical

When I previously posted about a model structure on Cat, I called the usual (categorical equivalence, injective-on-objects, isofibration) model structure the “standard model structure”. The nLab calls it the canonical model structure on Cat, but I dislike that name because it always seems to suggest that there is some mechanical procedure for constructing it. As it turns out – there is!

Recall that Rezk’s classifying diagram for a (small) category 𝒞 is the bisimplicial set (= simplicial simplicial set) defined by N(𝒞)n, m = Fun([n] × I[m], 𝒞), where [n] is the standard n-simplex in Cat and I is the groupoid completion functor (= left adjoint of the inclusion GrpdCat). Rezk [2001] has shown that N : CatssSet is fully faithful and is homotopically conservative in the sense of sending categorical equivalences to degreewise weak homotopy equivalences and reflecting degreewise weak homotopy equivalences as categorical equivalences. As is usual with presheaf categories, N has a left adjoint, namely the colimit-preserving functor τ1 : ssSetCat that sends the bisimplicial set representing [n, m] to the category [n] × I[m].

I claim that the standard model structure on Cat is the model structure obtained by transferring the projective model structure on ssSet (i.e. degreewise weak homotopy equivalences and degreewise Kan fibrations) along the left adjoint τ1 : ssSetCat. We have already remarked that N preserves and reflects weak equivalences, so it is enough to show that N also preserves and reflects fibrations. This is fairly straightforward.

The Morita model structure on Cat

There is a well-known model structure on Cat where the weak equivalences are the categorical equivalences, i.e. the functors that are fully faithful and essentially surjective on objects, the cofibrations are the functors that are injective on objects, and the fibrations are the isofibrations, i.e. the functors that lift isomorphisms. Let us say that a functor f : 𝒞 → 𝒟 is a Morita equivalence if the induced functor f* : [𝒟op, Set] → [𝒞op, Set] is a categorical equivalence. Clearly, every categorical equivalence is a Morita equivalence. Does the left Bousfield localisation of Cat with respect to Morita equivalences exist?

The standard model structure on Cat is combinatorial and simplicial, and all objects are cofibrant, so the model structure is also left proper. Thus, we may apply Smith’s theorem on the existence of left Bousfield localisations. It is a straightforward exercise to verify that left Bousfield localisation with respect to the inclusion of the free idempotent into the free split idempotent gives the desired model structure: the local objects are the Cauchy-complete categories, so the local equivalences are the Morita equivalences. (That Morita equivalences are local equivalences is easy; for the converse, consider a small category of sufficiently large sets.)

But one could (in principle) also establish the existence of the Morita model structure on Cat by hand: the fibrations are the isofibrations that also lift splittings of idempotents, i.e. the isofibrations with the right lifting property with respect to the inclusion mentioned above. Thus the Morita model structure on Cat is generated by the same cofibrations as the standard model structure on Cat and just one extra trivial cofibration.

You could have invented simplicial sets

Suppose you were an mid-20th-century algebraic topologist familiar with abstract simplicial complexes. What might lead you to invent simplicial sets? The historical answer, at least according to Eilenberg and Zilber [1950, Semi-simplicial complexes and singular homology], is that there are many important examples of such things: for instance, the “set” of singular simplices of a topological space is an example of such a thing. However, one could also motivate simplicial sets using more combinatorial considerations.

Recall the problem of triangulating the product of two simplicial complexes, say K and L. We can do this as follows:

  1. Choose a linear ordering of the vertices of K and L.
  2. The vertices of KL are pairs (x, y), where x is a vertex of K and y is a vertex of L.
  3. The n-simplices of KL are given by n + 1 distinct vertices, say { (x0, y0), …, (xn, yn) }, such that x0, …, xn and y0, …, yn are increasing sequences (but not necessarily all distinct), { x0, …, xn } is a simplex of K (possibly of dimension < n) and { y0, …, yn } is a simplex of L (possibly of dimension < n).

Unfortunately, KL need not have the universal property of a cartesian product of K and L unless we require simplicial maps to respect the linear ordering of the vertices at least “locally”. More precisely, we have to restrict attention to those simplicial maps that preserve the linear ordering of the vertices of each simplex of K and L; so if a pair of vertices do not appear in the same simplex, then the map need not preserve their relative ordering. In fact, we do not need to linearly order all the vertices in the first place – we only need the ordering “locally” – that is, a linear ordering of the vertices of each simplex, which is required to be compatible with the linear ordering of the vertices of its faces. Let us say a rigid simplicial complex is a simplicial complex equipped with such a “local” linear ordering, and a morphism of rigid simplicial complexes is a simplicial map that is “locally” order-preserving. (Note that there is a subtle difference between rigid simplicial complexes and ordered simplicial complexes: non-isomorphic ordered simplicial complexes can give rise to the same rigid simplicial complex. In this light, perhaps it would be better to say ‘semi-rigid’ instead of ‘rigid’.)

Clearly, there is a unique rigidification of the standard simplices up to isomorphism: all we have to do is choose a linear ordering of the vertices. It is a simple matter to describe the n-simplices of a rigid simplicial complex in terms of morphisms: there is a bijection between the set of n-simplices of K and the set of morphisms ΔnK that are injective on vertices, and it is natural so long as we restrict our attention to morphisms that are injective on vertices. Keeping in mind our construction of KL, this awkwardness suggests that the right thing to consider is the set of possibly-degenerate n-simplices of K, which we will identify with the set of all morphisms ΔnK. This is what the ‘complete’ in ‘complete semi-simplicial complex’ refers to.

It is at this point that one realises that one could reorganise the data of a rigid simplicial complex as a (many-sorted) algebraic structure such that KL is literally the cartesian product of K and L. Indeed, a morphism of rigid simplicial complexes is completely determined by its action on the sets of possibly-degenerate simplices, and in fact they are precisely the ones that respect the face/degeneracy relations. The fact that the vertices of each simplex are linearly ordered allows us to linearly order the faces of each simplex; thus one is led to the idea of face and degeneracy operators, and thence to the famous simplicial identities. This in turn enables us to identify morphisms of rigid simplicial complexes with “homomorphisms”, i.e. maps that commute with these operators.

In modern terms, what we have discovered here is a fully faithful embedding of the category of rigid simplicial complexes into the category of simplicial sets. There is one remaining question: how do we know whether a simplicial set is (isomorphic to) one that comes from a rigid simplicial complex? The answer turns out to be straightforward enough: the rigid simplicial complexes correspond to those simplicial sets whose simplices are “vertex-determined”, i.e. those X such that two (possibly-degenerate) n-simplices of X are equal if and only if their faces are equal. This condition (for a fixed n) is an example of what logicians call a Horn clause. So in some sense, the theory of simplicial sets is the purely equational fragment of the theory of rigid simplicial complexes, and this is one reason why the category of simplicial sets has better properties than the category of rigid simplicial complexes. The omission of the “vertex-determined” condition is what the ‘semi-’ in ‘(complete) semi-simplicial complex’ refers to.

The number of abstract simplicial complexes

On MathOverflow, François G. Dorais pointed out that the partially ordered set of abstract simplicial complexes with vertices drawn from the set { 0, …, n } is a free distributive lattice (with free top element but without free bottom element). This is not hard to see in hindsight: the generators are the (n – 1)-simplices. This poset is of interest to topos theorists because it is precisely the poset of simplicial subsets of the standard n-simplex; hence, we obtain a nice description of the subobject classifier of the topos of simplicial sets.

Amusingly, this observation implies it is very difficult to count the number of abstract simplicial complexes: apparently, the number of elements in a free distributive lattice is not known in general. This is Dedekind’s problem, and the known numbers are enumerated as OEIS sequence A000372.